# Cube subgroups

## Subgroups generated by face moves

Suppose not all possible cube moves are allowed, for example suppose that you only use moves of the Up and Right faces. You can mix a cube that way, and then try to solve it again using only those moves. Clearly not all cube positions can be reached, since the 2×3×3 block at the bottom left is never disturbed by the U and R moves. In the same way we might restrict ourselves to using only half turns. There are many choices as to which faces we allow, and whether each face uses only half turns or not. Below is a list containing all 56 possibilities.

Skip past table
Generators SizeCornersEdgesRestrictions
0. - 1= 1 * 1
1a.U2 2= 2 * 2 / 2
1b.U 4= 4 * 4 / 4
2a.U2, D2 4= 4 * 4 / 4
2b.U, D2 8= 8 * 8 / 8
2c.U, D 16=16 *16 /16
2d.U2, R2 12=3! *3! 22/2 /3!
2e.U, R2 14,400=6!/6 *5! 2! / 2
2f.U, R 73,483,200=6!/6 36/3*7! / 2
3a.U2, D2, F2 96=4! *4! 23/2 /4!
3b.U2, R2, F2 2,592=4! *3! 3! 3! / 2 / 1
3c.U, R2, F2 10,886,400=7! *6! 3! / 2
3d.U, R2, L2 58,060,800=8! *6! 2! 2! / 2
3e.U2, R, L2 58,060,800=8! *6! 2! 2! / 2
3f.U, R2, D 1,625,702,400=8! *8! 2! / 2
3g.U2, R, F 666,639,590,400=7! 37/3 *9! / 2
3h.U, R, D2 3,555,411,148,800=8! 38/3 *8! 2! / 2
3i.U, R, D 159,993,501,696,000=8! 38/3 *10! / 2
3j.U, R, F 170,659,735,142,400=7! 37/3 *9! 29/2 / 2
4a.U2, D2, R2, L2 192=4! *4! 24/2 /4!
4b.U2, D2, R2, F2 165,888=4! 4 *4! 4! 3! / 2 / 1
4c.U, D2, R2, L2 116,121,600=8! *6! 2! 2! 2! / 2
4d.U2, D2, R, F2 2,438,553,600=8! *7! 4! / 2
4f.U, D2, R2, F2 4,877,107,200=8! *8! 3! / 2
4e.U, D, R2, L2 3,251,404,800=8! *8! 2! 2! / 2
4g.U, D, R2, F2 4,877,107,200=8! *8! 3! / 2
4h.U, D2, R, F2 1,759,928,518,656,000=8! 38/3 *11! / 2
4i.U, D, R, F2 1,759,928,518,656,000=8! 38/3 *11! / 2
4j.U2, D2, R, F 1,759,928,518,656,000=8! 38/3 *11! / 2
4k.U, D2, R, L2 7,110,822,297,600=8! 38/3 *8! 2! 2! / 2
4l.U, D2, F, R 1,802,166,803,103,744,000=8! 38/3 *11! 211/2/ 2
4m.U, D2, R, L 319,987,003,392,000=8! 38/3 *10! 2! / 2
4n.U, D, F, R 1,802,166,803,103,744,000=8! 38/3 *11! 211/2/ 2
4o.U, D, R, L 21,119,142,223,872,000=8! 38/3 *12! / 2
5a.U2, F2, B2, R2, L2 663,552=4! 4 *4! 4! 4! / 2 / 1
5b.U, D2, F2, B2, R2 19,508,428,800=8! *8! 4! / 2
5c.U, D, F2, B2, R2 19,508,428,800=8! *8! 4! / 2
5d.U, F2, B2, R2, L2 19,508,428,800=8! *8! 4! / 2
5e.U2, F, B2, R, L2 21,119,142,223,872,000=8! 38/3 *12! / 2
5f.U2, F, B2, R, L 21,119,142,223,872,000=8! 38/3 *12! / 2
5g.U2, F, B, R, L 21,119,142,223,872,000=8! 38/3 *12! / 2
5h.U2, D2, F, B, R 21,119,142,223,872,000=8! 38/3 *12! / 2
5i.U2, D2, F, B2, R 21,119,142,223,872,000=8! 38/3 *12! / 2
5j.U, F, B2, R, L2 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2
5k.U, F, B, R, L2 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2
5l.U, F, B, R, L 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2
6a.U2, D2, F2, B2, R2, L2 663,552=4! 4 *4! 4! 4! / 2 / 1
6b.U, D, F2, B2, R2, L2 19,508,428,800=8! *8! 4! / 2
6c.U, D2, F2, B2, R2, L2 19,508,428,800=8! *8! 4! / 2
6d.U2, D2, F, B, R, L 21,119,142,223,872,000=8! 38/3 *12! / 2
6e.U2, D2, F, B2, R, L 21,119,142,223,872,000=8! 38/3 *12! / 2
6f.U2, D2, F, B2, R, L2 21,119,142,223,872,000=8! 38/3 *12! / 2
6g.U, D2, F, B2, R, L2 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2
6h.U, D2, F, B, R, L2 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2
6i.U, D2, F, B, R, L 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2
6j.U, D, F, B, R, L 43,252,003,274,489,856,000=8! 38/3 *12! 212/2/ 2

The list above is complete up to turns of the whole cube, e.g. <F,R> is not included because it is isomorphic to the group generated by U and R, which is on the list.

The number of reachable positions (the size of the group) is shown, as well as the number of ways the corners or the edges can be arranged separately. The last column is a factor showing how much the edges and corners restrict each other. Usually this is a factor 2, due to the normal parity restriction that the parity of the total corner and edge permutation must be even. Thus once the corners have been solved the parity of the edges is forced, and they have half as many possible arrangements than they had before the corners were solved. In some rare cases the edges and corners influence each other by more than this.

Some of the groups listed are actually identical. This can easily be shown using the following three move sequences.

 1 D = F2R2D2F2U2R2F2 U F2R2U2F2D2R2F2 ( uses U, F2, R2, D2 ) 2 D2 = F2R2L2B2 U2 F2R2L2B2 ( uses F2, B2, R2, L2, U2 ) 3 U2 = FR' FLFL' F2R2 B'RBR F'R ( uses F, B, R, L )

For example 4f=<U, D2, F2, R2> and 4g=<U, D, F2, R2> are the same because with the first move sequence you can use only U, D2, F2, and R2 to get the same effect as D (and conversely with D you can of course get D2). Similarly 4h, 4i are equal. In fact, using these two move sequences you can show that any two groups on the list of the same size are identical groups, except that 4j is slightly different from 4h/4i. In group 4j the edge orientations differ from those in 4h/4i, though in all other respects they are the same. Thus they are isomorphic groups.

If you wish to solve a position using the same set of moves you used to mix them, it is often easiest to solve the corners first, and then the edges. Below is a table of useful move sequences. With conjugates of these, all positions in the above groups can be solved. Note that the sequences for the corners may disturb edges.

URF- URB+ U, R RU'RU'RU' R'UR'UR'U
(UFL, DFR, DBR) U, R2, F2 UF2U'R2UF2U'F2
(URF, DRB, DBL) U, R2, L2 UL2U'R2 UL2U'R2
(URF, DRB, DBL) U2, R, L2 R2U2R'L2U2R'L2U2RL2U2RL2U2
UF+ UR+ U, R, F RU'R2UFRUF2U'FR2F2R2F2
(UF, UR, UB) U, R RU2RURUR2U'R'U'R2
(UF, UR, UB) U, R2 U2R2U2R2 UR2UR2 U2R2U2R2 UR2UR2
(UF, UB)(RF, RB)U2, R2 R2U2 R2U2 R2U2
(FL, FR, BR) U2, R2, F2F2U2R2U2 F2U2R2U2

## Subgroups generated by moves of middle slices and faces

In the previous section only face turns were used and so the face centres remained fixed. In this section the generators will include moves of one or more of the middle layers. I will only list those groups for which there is at least one corner or edge that remains in place that can serve as a reference point when solving the puzzle. This leads to 272 more ways to generate a group.

Skip past table
GeneratorsFixed pieces SizeCornersEdgesCentresRestrictions
1a.Um2 U and D layers 2= 1* 2* 2/ 2
1b.Um U and D layers 4= 1* 4* 4/ 4
2a.U2, Um2 D layer 4= 2* 2 2* 2/ 2
2b.U2, Um D layer 8= 2* 2 4* 4/ 8
2c.U, Um2 D layer 8= 4* 2 4* 2/ 8
2d.U, Um D layer 16= 4* 4 4* 4/ 16
2e.U2, Rm2 DBL BL DL L DFL FL DBR BR DR R DFR FR 8= 2* 8* 2/ 4
2f.U2, Rm DBL BL DL L DFL FL DBR BR DR R DFR FR 96= 2* 2! 4!* 4/ 4
2g.U, Rm2 DBL BL DL L DFL FL DBR BR DR R DFR FR 96= 4* 3! 23/2* 2/ 4
2h.U, Rm DBL BL DL L DFL FL DBR BR DR R DFR FR 184,320= 4* 6! 26/2* 4/ 2
2i.Um2, Rm2 UL UR DL DR and corners 4= 1* 2 2*2 2/ 4
2j.Um, Rm2 UL UR DL DR and corners 16= 1* 2 4*2 4/ 4
2k.Um, Rm UL UR DL DR and corners 192= 1* 4 4*24/ 2
3a.U2, D2, Rm2 BL L FL BR R FR 16= 2 2* 2 2 4* 2/ 8
3b.U2, D, Rm2 BL L FL BR R FR 192= 2 4* 2 3! 23* 2/ 8
3c.U2, D2, Rm BL L FL BR R FR 192= 2 2* 2 2 4!* 4/ 8
3d.U2, D, Rm BL L FL BR R FR 368,640= 2 4* 2! 6! 26/2* 4/ 4
3e.U, D, Rm2 BL L FL BR R FR 3072= 4 4* 4! 24* 2/ 4
3f.U, D, Rm BL L FL BR R FR 165,150,720= 4 4* 8! 28/2* 4/ 2
3g.U2, R2, Um2 DBL DB DL D DFL DF 48= 3!* 3! 23* 2/12
3h.U, R2, Um2 DBL DB DL D DFL DF 57,600= 6!/6* 8 5!* 2/ 4
3i.U2, R, Um2 DBL DB DL D DFL DF 1,209,600= 6!/6* 2 7!* 2/ 2
3j.U2, R2, Um DBL DB DL D DFL DF 576= 3!* 2! 3! 4!* 4/12
3k.U, R, Um2 DBL DB DL D DFL DF 10,581,580,800= 6!/6 36/3* 9!* 2/ 2
3l.U, R2, Um DBL DB DL D DFL DF 691,200= 6!/6* 2 4! 5!* 4/ 4
3m.U2, R, Um DBL DB DL D DFL DF 154,828,800= 6!/6* 2! 7! 27/2* 4/ 2
3n.U, R, Um DBL DB DL D DFL DF 5,417,769,369,600= 6!/6 36/3* 9! 29/2* 4/ 2
3o.U2, R2, Fm2 DBL BL DB B DFL FL DF F 96= 3!* 2 4!* 2/ 6
3p.U, R2, Fm2 DBL BL DB B DFL FL DF F 172,800= 6!/6* 2 6!* 2/ 2
3q.U2, R2, Fm DBL BL DB B DFL FL DF F 576= 3!* 2! 2! 4!* 4/ 4
3r.U, R, Fm2 DBL BL DB B DFL FL DF F 1,175,731,200= 6!/6 36/3* 8!* 2/ 2
3s.U, R2, Fm DBL BL DB B DFL FL DF F 11,059,200= 6!/6* 2! 6! 26/2* 4/ 2
3t.U, R, Fm DBL BL DB B DFL FL DF F 300,987,187,200= 6!/6 36/3* 8! 28/2* 4/ 2
3u.U2, Um2, Rm2 DBL DL DFL DBR DR DFR 16= 2* 8 2!*2 2/ 8
3v.U, Um2, Rm2 DBL DL DFL DBR DR DFR 192= 4* 2 3! 23*2 2/ 8
3w.U2, Um, Rm2 DBL DL DFL DBR DR DFR 64= 2* 4 8*2 4/ 8
3x.U2, Um2, Rm DBL DL DFL DBR DR DFR 192= 2* 2 2 4!*2 4/ 8
3y.U, Um, Rm2 DBL DL DFL DBR DR DFR 768= 4* 4 3! 23*2 4/ 8
3z.U, Um2, Rm DBL DL DFL DBR DR DFR 368,640= 4* 2! 6! 26/2*2 4/ 4
3aa.U2, Um, Rm DBL DL DFL DBR DR DFR 2,304= 2* 4 2 4!*24/ 4
3ab.U, Um, Rm DBL DL DFL DBR DR DFR 4,423,680= 4* 4 6! 26/2*24/ 2
3ac.U2, Rm2, Fm2 DBL BL DBR BR DFL FL DFR FR 32= 2* 8 8 / 2*2 2/ 8
3ad.U, Rm2, Fm2 DBL BL DBR BR DFL FL DFR FR 1,536= 4* 4! 24*2 2/ 4
3ae.U2, Rm, Fm2 DBL BL DBR BR DFL FL DFR FR 384= 2* 8 4!*2 4/ 8
3af.U, Rm, Fm2 DBL BL DBR BR DFL FL DFR FR 82,575,360= 4* 8! 28/2*2 4/ 2
3ag.U2, Rm, Fm DBL BL DBR BR DFL FL DFR FR 13,824= 2* 2 4! 4!*24/ 4
3ah.U, Rm, Fm DBL BL DBR BR DFL FL DFR FR 247,726,080= 4* 8! 28/2*24/ 2
3ai.Um2, Rm2, Fm2 All corners 8= 1* 2 2 2*2 2 2 / 2/ 4
3aj.Um, Rm2, Fm2 All corners 32= 1* 2 2 4*2 4/ 4
3ak.Um, Rm, Fm2 All corners 384= 1* 2 4 4*24/ 2
3al.Um, Rm, Fm All corners 768= 1* 4 4 4*24/ 2
4a.U2, F2, R2, Um2 DBL DL DB D 20,736= 4!* 3! 3! 3! / 2* 2/ 1
4b.U, F2, R2, Um2 DBL DL DB D 87,091,200= 7!* 6! 4!* 2/ 2
4c.U2, F2, R2, Um DBL DL DB D 41,472= 4!* 4! 3! 3!* 4/ 2
4d.U2, F, R2, Um2 DBL DL DB D 152,409,600= 7!* 3! 7!* 2/ 2
4e.U, F, R2, Um2 DBL DL DB D 13,332,791,808,000= 7! 37/3* 10!* 2/ 2
4f.U2, F, R, Um2 DBL DL DB D 13,332,791,808,000= 7! 37/3* 10!* 2/ 2
4g.U, F2, R2, Um DBL DL DB D 174,182,400= 7!* 2 4! 6!* 4/ 4
4h.U2, F2, R, Um DBL DL DB D 19,508,428,800= 7!* 2 4! 8!* 4/ 2
4i.U, F, R, Um2 DBL DL DB D 6,826,389,405,696,000= 7! 37/3* 10! 210/2* 2/ 2
4j.U, F, R2, Um DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
4k.U2, F, R, Um DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
4l.U, F, R, Um DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
4m.U2, R2, D2, Rm2 BL L FL 192= 4!* 8 4!* 2/48
4n.U, R2, D2, Rm2 BL L FL 3,251,404,800= 8!* 2 8!* 2/ 2
4o.U2, R, D2, Rm2 BL L FL 116,121,600= 8!* 8 6!* 2/ 4
4p.U2, R2, D2, Rm BL L FL 2,304= 4!* 2 4! 4!* 4/48
4q.U, R, D2, Rm2 BL L FL 319,987,003,392,000= 8! 38/3* 10!* 2/ 2
4r.U, R2, D, Rm2 BL L FL 3,251,404,800= 8!* 2 8!* 2/ 2
4s.U, R2, D2, Rm BL L FL 832,359,628,800= 8!* 2 8! 28/2* 4/ 2
4t.U2, R, D2, Rm BL L FL 1,393,459,200= 8!* 2 4! 6!* 4/ 2
4u.U, R, D, Rm2 BL L FL 319,987,003,392,000= 8! 38/3* 10!* 2/ 2
4v.U, R, D2, Rm BL L FL 327,666,691,473,408,000= 8! 38/3* 10! 210/2* 4/ 2
4w.U, R2, D, Rm BL L FL 832,359,628,800= 8!* 2 8! 28/2* 4/ 2
4x.U, R, D, Rm BL L FL 327,666,691,473,408,000= 8! 38/3* 10! 210/2* 4/ 2
4y.U2, R2, D2, Fm2 BL B FL F 768= 4!* 2! 2! 2! 4! / 2* 2/ 6
4z.U, R2, D2, Fm2 BL B FL F 116,121,600= 8!* 2! 2! 6!* 2/ 2
4aa.U2, R, D2, Fm2 BL B FL F 116,121,600= 8!* 2! 2! 6!* 2/ 2
4ab.U2, R2, D2, Fm BL B FL F 4,608= 4!* 2! 2! 2! 4!* 4/ 4
4ac.U, R, D2, Fm2 BL B FL F 7,110,822,297,600= 8! 38/3* 2 8!* 2/ 2
4ad.U, R2, D, Fm2 BL B FL F 3,251,404,800= 8!* 2 8!* 2/ 2
4ae.U, R2, D2, Fm BL B FL F 7,431,782,400= 8!* 2! 2! 6! 26/2* 4/ 2
4af.U2, R, D2, Fm BL B FL F 7,431,782,400= 8!* 2! 2! 6! 26/2* 4/ 2
4ag.U, R, D, Fm2 BL B FL F 319,987,003,392,000= 8! 38/3* 10!* 2/ 2
4ah.U, R, D2, Fm BL B FL F 1,820,370,508,185,600= 8! 38/3* 2 8! 28/2* 4/ 2
4ai.U, R2, D, Fm BL B FL F 832,359,628,800= 8!* 2 8! 28/2* 4/ 2
4aj.U, R, D, Fm BL B FL F 327,666,691,473,408,000= 8! 38/3* 10! 210/2* 4/ 2
4ak.U2, R2, Um2, Rm2 DBL DL DFL 192= 3!* 8 8 3! / 2*2 2/24
4al.U, R2, Um2, Rm2 DBL DL DFL 4,838,400= 6!/6* 8!*2 2/ 4
4am.U2, R2, Um, Rm2 DBL DL DFL 2,304= 3!* 8 3! 4!*2 4/24
4an.U, R, Um2, Rm2 DBL DL DFL 2,327,947,776,000= 6!/6 36/3* 11!*2 2/ 2
4ao.U, R2, Um, Rm2 DBL DL DFL 58,060,800= 6!/6* 2 4! 7!*2 4/ 4
4ap.U2, R, Um, Rm2 DBL DL DFL 619,315,200= 6!/6* 8 7! 27/2*2 4/ 4
4aq.U2, R2, Um, Rm DBL DL DFL 82,944= 3!* 2 4! 4! 3!*24/12
4ar.U, R, Um, Rm2 DBL DL DFL 4,767,637,045,248,000= 6!/6 36/3* 11! 211/2*2 4/ 2
4as.U, R2, Um, Rm DBL DL DFL 22,295,347,200= 6!/6* 4! 2 7! 27/2*24/ 2
4at.U, R, Um, Rm DBL DL DFL 14,302,911,135,744,000= 6!/6 36/3* 11! 211/2*24/ 2
4au.U2, F2, Um2, Rm2 DBL BL DBR BR 384= 3!* 8 2! 4! / 2*2 2/12
4av.U, F2, Um2, Rm2 DBL BL DBR BR 691,200= 6!/6* 8 6!*2 2/ 4
4aw.U2, F, Um2, Rm2 DBL BL DBR BR 19,353,600= 6!/6* 2 8!*2 2/ 2
4ax.U2, F2, Um, Rm2 DBL BL DBR BR 4,608= 3!* 2 4! 4!*2 4/12
4ay.U2, F2, Um2, Rm DBL BL DBR BR 2,304= 3!* 8 2! 4!*2 4/ 8
4az.U, F, Um2, Rm2 DBL BL DBR BR 211,631,616,000= 6!/6 36/3* 10!*2 2/ 2
4ba.U, F2, Um, Rm2 DBL BL DBR BR 8,294,400= 6!/6* 2 4! 6!*2 4/ 4
4bb.U2, F, Um, Rm2 DBL BL DBR BR 4,954,521,600= 6!/6* 2 8! 28/2*2 4/ 2
4bc.U, F2, Um2, Rm DBL BL DBR BR 44,236,800= 6!/6* 8 6! 26/2*2 4/ 4
4bd.U2, F, Um2, Rm DBL BL DBR BR 4,954,521,600= 6!/6* 2 8! 28/2*2 4/ 2
4be.U2, F2, Um, Rm DBL BL DBR BR 82,944= 3!* 2! 2! 4! 4!*24/ 4
4bf.U, F, Um, Rm2 DBL BL DBR BR 216,710,774,784,000= 6!/6 36/3* 10! 210/2*2 4/ 2
4bg.U, F, Um2, Rm DBL BL DBR BR 216,710,774,784,000= 6!/6 36/3* 10! 210/2*2 4/ 2
4bh.U, F2, Um, Rm DBL BL DBR BR 1,592,524,800= 6!/6* 2 4! 6! 26/2*24/ 2
4bi.U2, F, Um, Rm DBL BL DBR BR 14,863,564,800= 6!/6* 2 8! 28/2*24/ 2
4bj.U, F, Um, Rm DBL BL DBR BR 650,132,324,352,000= 6!/6 36/3* 10! 210/2*24/ 2
4bk.U2, D2, Rm2, Fm2 BL BR FL FR 64= 2 2* 8 8 / 2*2 2/ 8
4bl.U, D2, Rm2, Fm2 BL BR FL FR 3,072= 2 4* 4! 24*2 2/ 4
4bm.U2, D2, Rm, Fm2 BL BR FL FR 768= 2 2* 8 4!*2 4/ 8
4bn.U, D, Rm2, Fm2 BL BR FL FR 6,144= 4 4* 4! 24*2 2/ 4
4bo.U, D2, Rm, Fm2 BL BR FL FR 165,150,720= 2 4* 8! 28/2*2 4/ 2
4bp.U2, D2, Rm, Fm BL BR FL FR 27,648= 2 2* 2 4! 4!*24/ 4
4bq.U, D, Rm, Fm2 BL BR FL FR 330,301,440= 4 4* 8! 28/2*2 4/ 2
4br.U, D2, Rm, Fm BL BR FL FR 495,452,160= 2 4* 8! 28/2*24/ 2
4bs.U, D, Rm, Fm BL BR FL FR 990,904,320= 4 4* 8! 28/2*24/ 2
4bt.U2, Um2, Rm2, Fm2 DBL DBR DFL DFR 64= 2* 2 8 8 / 2*2 2 2 / 2/ 8
4bu.U, Um2, Rm2, Fm2 DBL DBR DFL DFR 3,072= 4* 2 4! 24*2 2 2 / 2/ 4
4bv.U2, Um, Rm2, Fm2 DBL DBR DFL DFR 256= 2* 4 8 8 / 2*2 4/ 8
4bw.U2, Um2, Rm, Fm2 DBL DBR DFL DFR 768= 2* 2 8 4!*2 4/ 8
4bx.U, Um, Rm2, Fm2 DBL DBR DFL DFR 6,144= 4* 4 4! 24*2 4/ 8
4by.U, Um2, Rm, Fm2 DBL DBR DFL DFR 165,150,720= 4* 4 8! 28/2*2 4/ 2
4bz.U2, Um, Rm, Fm2 DBL DBR DFL DFR 9,216= 2* 4 8 4!*24 / 4
4ca.U2, Um2, Rm, Fm DBL DBR DFL DFR 27,648= 2* 2 4! 4! 2*24 / 4
4cb.U, Um, Rm, Fm2 DBL DBR DFL DFR 990,904,320= 4* 4 8! 28/2*24 / 2
4cc.U, Um2, Rm, Fm DBL DBR DFL DFR 495,452,160= 4* 2 8! 28/2*24 / 2
4cd.U2, Um, Rm, Fm DBL DBR DFL DFR 55,296= 2* 2 4! 4! 4*24 / 4
4ce.U, Um, Rm, Fm DBL DBR DFL DFR 990,904,320= 4* 4 8! 28/2*24 / 2
5a.U2, F2, R2, L2, Um2 D DB 331,776= 4! 4!/6* 3! 4! 4! / 2* 2/ 1
5b.U, F2, R2, L2, Um2 D DB 4,877,107,200= 8!* 4! 7!* 2/ 2
5c.U2, F, R2, L2, Um2 D DB 4,877,107,200= 8!* 4! 7!* 2/ 2
5d.U2, F2, R, L2, Um2 D DB 9,754,214,400= 8!* 3! 8!* 2/ 2
5e.U2, F2, R2, L2, Um D DB 663,552= 4! 4!/6* 3! 4! 4!* 4/ 2
5f.U, F, R2, L2, Um2 D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
5g.U, F2, R, L2, Um2 D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
5h.U2, F, R, L2, Um2 D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
5i.U2, F2, R, L, Um2 D DB 9,754,214,400= 8!* 3! 8!* 2/ 2
5j.U, F2, R2, L2, Um D DB 9,754,214,400= 8!* 2 4! 7!* 4/ 4
5k.U2, F, R2, L2, Um D DB 624,269,721,600= 8!* 4! 7! 27/2* 4/ 2
5l.U2, F2, R, L2, Um D DB 2,497,078,886,400= 8!* 3! 8! 28/2* 4/ 2
5m.U, F, R, L2, Um2 D DB 3,604,333,606,207,488,000= 8! 38/3* 11! 211/2* 2/ 2
5n.U, F2, R, L, Um2 D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
5o.U2, F, R, L, Um2 D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
5p.U, F, R2, L2, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5q.U, F2, R, L2, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5r.U2, F, R, L2, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5s.U2, F2, R, L, Um D DB 2497,078,886,400= 8!* 3! 8! 28/2* 4/ 2
5t.U, F, R, L, Um2 D DB 3,604,333,606,207,488,000= 8! 38/3* 11! 211/2* 2/ 2
5u.U, F, R, L2, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5v.U, F2, R, L, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5w.U2, F, R, L, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5x.U, F, R, L, Um D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5y.U2, R2, D2, Rm2, Fm2 BL FL 1,536= 4!* 8 4! / 2*2 2/ 12
5z.U, R2, D2, Rm2, Fm2 BL FL 6,502,809,600= 8!* 2 8!*2 2/ 2
5aa.U2, R, D2, Rm2, Fm2 BL FL 232,243,200= 8!* 2 2! 2! 6!*2 2/ 4
5ab.U2, R2, D2, Rm, Fm2 BL FL 18,432= 4!* 2 4! 4!*2 4/ 12
5ac.U2, R2, D2, Rm2, Fm BL FL 9,216= 4!* 2 4! 8*2 4/ 8
5ad.U, R, D2, Rm2, Fm2 BL FL 639,974,006,784,000= 8! 38/3* 10!*2 2/ 2
5ae.U, R2, D, Rm2, Fm2 BL FL 6,502,809,600= 8!* 2 8!*2 2/ 2
5af.U, R2, D2, Rm, Fm2 BL FL 1664,719,257,600= 8!* 2 8! 28/2*2 4/ 2
5ag.U2, R, D2, Rm, Fm2 BL FL 2,786,918,400= 8!* 2 4! 6!*2 4/ 4
5ah.U, R2, D2, Rm2, Fm BL FL 1664,719,257,600= 8!* 2 8! 28/2*2 4/ 2
5ai.U2, R, D2, Rm2, Fm BL FL 14,863,564,800= 8!* 8 6! 26/2*2 4/ 4
5aj.U2, R2, D2, Rm, Fm BL FL 331,776= 4!* 2 4! 2 4!*24/ 4
5ak.U, R, D, Rm2, Fm2 BL FL 639,974,006,784,000= 8! 38/3* 10!*2 2/ 2
5al.U, R, D2, Rm, Fm2 BL FL 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
5am.U, R2, D, Rm, Fm2 BL FL 1,664,719,257,600= 8!* 2 8! 28/2*2 4/ 2
5an.U, R, D2, Rm2, Fm BL FL 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
5ao.U, R2, D, Rm2, Fm BL FL 1,664,719,257,600= 8!* 2 8! 28/2*2 4/ 2
5ap.U, R2, D2, Rm, Fm BL FL 4,994,157,772,800= 8!* 2 8! 28/2*24/ 2
5aq.U2, R, D2, Rm, Fm BL FL 535,088,332,800= 8!* 2 4! 6! 26/2*24/ 2
5ar.U, R, D, Rm, Fm2 BL FL 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
5as.U, R, D, Rm2, Fm BL FL 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
5at.U, R, D2, Rm, Fm BL FL 1,966,000,148,840,448,000= 8! 38/3* 10! 210/2*24/ 2
5au.U, R2, D, Rm, Fm BL FL 4,994,157,772,800= 8!* 2 8! 28/2*24/ 2
5av.U, R, D, Rm, Fm BL FL 1,966,000,148,840,448,000= 8! 38/3* 10! 210/2*24/ 2
5aw.U2, R2, F2, Rm2, Fm2 DBL BL 165,888= 4!* 3! 4! 4! / 2*2 2/ 1
5ax.U, R2, F2, Rm2, Fm2 DBL BL 2,438,553,600= 7!* 3! 8!*2 2/ 2
5ay.U2, R, F2, Rm2, Fm2 DBL BL 1,219,276,800= 7!* 4! 7!*2 2/ 2
5az.U2, R2, F2, Rm, Fm2 DBL BL 331,776= 4!* 3! 4! 4!*2 4/ 2
5ba.U, R, F2, Rm2, Fm2 DBL BL 293,321,419,776,000= 7! 37/3* 11!*2 2/ 2
5bb.U2, R, F, Rm2, Fm2 DBL BL 293,321,419,776,000= 7! 37/3* 11!*2 2/ 2
5bc.U, R2, F2, Rm, Fm2 DBL BL 624,269,721,600= 7!* 3! 8! 28/2*2 4/ 2
5bd.U2, R, F2, Rm, Fm2 DBL BL 2,438,553,600= 7!* 2 4! 7!*2 4/ 4
5be.U2, R2, F, Rm, Fm2 DBL BL 156,067,430,400= 7!* 4! 7! 27/2*2 4/ 2
5bf.U2, R2, F2, Rm, Fm DBL BL 1,990,656= 4!* 2 4! 4! 3!*24/ 2
5bg.U, R, F, Rm2, Fm2 DBL BL 300,361,133,850,624,000= 7! 37/3* 11! 211/2*2 2/ 2
5bh.U, R, F2, Rm, Fm2 DBL BL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5bi.U, R2, F, Rm, Fm2 DBL BL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5bj.U2, R, F, Rm, Fm2 DBL BL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5bk.U, R2, F2, Rm, Fm DBL BL 1,872,809,164,800= 7!* 3! 8! 28/2*24/ 2
5bl.U2, R, F2, Rm, Fm DBL BL 936,404,582,400= 7!* 4! 2 7! 27/2*24/ 2
5bm.U, R, F, Rm, Fm2 DBL BL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5bn.U, R, F2, Rm, Fm DBL BL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
5bo.U2, R, F, Rm, Fm DBL BL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
5bp.U, R, F, Rm, Fm DBL BL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
5bq.U2, R2, Um2, Rm2, Fm2 DBL DFL 1,536= 3!* 8 8 4! / 2*2 2 2 / 2/ 12
5br.U, R2, Um2, Rm2, Fm2 DBL DFL 77,414,400= 6!/6* 8 8!*2 2 2 / 2/ 2
5bs.U2, R2, Um, Rm2, Fm2 DBL DFL 18,432= 3!* 2 4 4! 4!*2 4/ 12
5bt.U2, R2, Um2, Rm2, Fm DBL DFL 9,216= 3!* 8 8 4!*2 4/ 8
5bu.U, R, Um2, Rm2, Fm2 DBL DFL 27,935,373,312,000= 6!/6 36/3* 12!*2 2 2 / 2/ 2
5bv.U, R2, Um, Rm2, Fm2 DBL DFL 464,486,400= 6!/6* 2 4! 8!*2 4/ 4
5bw.U, R2, Um2, Rm, Fm2 DBL DFL 19,818,086,400= 6!/6* 8 8! 28/2*2 4/ 2
5bx.U, R2, Um2, Rm2, Fm DBL DFL 19,818,086,400= 6!/6* 8 8! 28/2*2 4/ 2
5by.U2, R2, Um, Rm, Fm2 DBL DFL 331,776= 3!* 2 4! 4! 4!*24 / 12
5bz.U2, R2, Um, Rm2, Fm DBL DFL 331,776= 3!* 8 2 4! 4!*24 / 4
5ca.U, R, Um, Rm2, Fm2 DBL DFL 114,423,289,085,952,000= 6!/6 36/3* 12! 212/2*2 4/ 2
5cb.U, R, Um2, Rm2, Fm DBL DFL 114,423,289,085,952,000= 6!/6 36/3* 12! 212/2*2 4/ 2
5cc.U, R2, Um, Rm, Fm2 DBL DFL 356,725,555,200= 6!/6* 4! 2 8! 28/2*24 / 2
5cd.U, R2, Um, Rm2, Fm DBL DFL 356,725,555,200= 6!/6* 4! 2 8! 28/2*24 / 2
5ce.U, R2, Um2, Rm, Fm DBL DFL 59,454,259,200= 6!/6* 8 8! 28/2*24 / 2
5cf.U2, R2, Um, Rm, Fm DBL DFL 1,990,656= 3!* 2 4! 2 4! 4!*24 / 4
5cg.U, R, Um, Rm, Fm2 DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24 / 2
5ch.U, R, Um, Rm2, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24 / 2
5ci.U, R2, Um, Rm, Fm DBL DFL 356,725,555,200= 6!/6* 4! 2 8! 28/2*24 / 2
5cj.U, R, Um, Rm, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24 / 2
6a.U2, F2, R2, D2, Rm2, Fm2 BL 663,552= 4! 4!/6* 3! 4! 4! / 2*2 2/ 1
6b.U, F2, R2, D2, Rm2, Fm2 BL 19,508,428,800= 8!* 3! 8!*2 2/ 2
6c.U2, F, R2, D2, Rm2, Fm2 BL 9,754,214,400= 8!* 4! 7!*2 2/ 2
6d.U2, F2, R2, D2, Rm, Fm2 BL 1,327,104= 4! 4!/6* 3! 4! 4!*2 4/ 2
6e.U, F, R2, D2, Rm2, Fm2 BL 7,039,714,074,624,000= 8! 38/3* 11!*2 2/ 2
6f.U2, F, R, D2, Rm2, Fm2 BL 7,039,714,074,624,000= 8! 38/3* 11!*2 2/ 2
6g.U, F2, R2, D, Rm2, Fm2 BL 19,508,428,800= 8!* 3! 8!*2 2/ 2
6h.U, F2, R2, D2, Rm, Fm2 BL 4,994,157,772,800= 8!* 3! 8! 28/2*2 4/ 2
6i.U2, F, R2, D2, Rm, Fm2 BL 1,248,539,443,200= 8!* 4! 7! 27/2*2 4/ 2
6j.U2, F2, R, D2, Rm, Fm2 BL 19,508,428,800= 8!* 2 4! 7!*2 4/ 4
6k.U2, F2, R2, D2, Rm, Fm BL 7,962,624= 4! 4!/6* 2 4! 4! 3!*24/ 2
6l.U, F, R, D2, Rm2, Fm2 BL 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2*2 2/ 2
6m.U, F, R2, D, Rm2, Fm2 BL 7,039,714,074,624,000= 8! 38/3* 11!*2 2/ 2
6n.U, F, R2, D2, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6o.U, F2, R, D2, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6p.U2, F, R, D2, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6q.U, F2, R2, D, Rm, Fm2 BL 4,994,157,772,800= 8!* 3! 8! 28/2*2 4/ 2
6r.U, F2, R2, D2, Rm, Fm BL 14,982,473,318,400= 8!* 3! 8! 28/2*24/ 2
6s.U2, F, R2, D2, Rm, Fm BL 7,491,236,659,200= 8!* 4! 2 7! 27/2*24/ 2
6t.U, F, R, D, Rm2, Fm2 BL 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2*2 2/ 2
6u.U, F, R, D2, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6v.U, F, R2, D, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6w.U, F2, R, D, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6x.U, F, R2, D2, Rm, Fm BL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
6y.U2, F, R, D2, Rm, Fm BL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
6z.U, F2, R2, D, Rm, Fm BL 14,982,473,318,400= 8!* 3! 8! 28/2*24/ 2
6aa.U, F, R, D, Rm, Fm2 BL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6ab.U, F, R, D2, Rm, Fm BL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
6ac.U, F, R2, D, Rm, Fm BL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
6ad.U, F, R, D, Rm, Fm BL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
6ae.U2, R2, F2, Um2, Rm2, Fm2DBL 663,552= 4!* 2 4! 4! 3!*2 2 2 / 2/ 1
6af.U, R2, F2, Um2, Rm2, Fm2 DBL 9,754,214,400= 7!* 4! 8!*2 2 2 / 2/ 2
6ag.U2, R2, F2, Um, Rm2, Fm2 DBL 1,327,104= 4!* 4! 4! 4!*2 4/ 2
6ah.U, R, F2, Um2, Rm2, Fm2 DBL 3,519,857,037,312,000= 7! 37/3* 12!*2 2 2 / 2/ 2
6ai.U, R2, F2, Um, Rm2, Fm2 DBL 19,508,428,800= 7!* 2 4! 8!*2 4/ 4
6aj.U, R2, F2, Um2, Rm, Fm2 DBL 2,497,078,886,400= 7!* 4! 8! 28/2*2 4/ 2
6ak.U2, R2, F2, Um, Rm, Fm2 DBL 7,962,624= 4!* 2 4! 2 4! 4! / 2*24/ 2
6al.U, R, F, Um2, Rm2, Fm2 DBL 7,208,667,212,414,976,000= 7! 37/3* 12! 212/2*2 2 2 / 2/ 2
6am.U, R, F2, Um, Rm2, Fm2 DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
6an.U2, R, F, Um, Rm2, Fm2 DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
6ao.U, R2, F2, Um, Rm, Fm2 DBL 14,982,473,318,400= 7!* 4! 2 8! 28/2*24/ 2
6ap.U2, R2, F, Um, Rm, Fm2 DBL 7,491,236,659,200= 7!* 4! 8! 28/2*24/ 2
6aq.U2, R2, F2, Um, Rm, Fm DBL 15,925,248= 4!* 2 4! 2 4! 4!*24/ 2
6ar.U, R, F, Um, Rm2, Fm2 DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
6as.U, R, F2, Um, Rm, Fm2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
6at.U, R2, F, Um, Rm, Fm2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
6au.U, R2, F2, Um, Rm, Fm DBL 14,982,473,318,400= 7!* 4! 2 8! 28/2*24/ 2
6av.U, R, F, Um, Rm, Fm2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
6aw.U, R, F2, Um, Rm, Fm DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
6ax.U, R, F, Um, Rm, Fm DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2

Some of the groups on this list have in a way already been encountered on the first list. For example the square group <U2, D2, F2, B2, R2, L2> is similar to the groups <U2, F2, R2, D2, Rm2, Fm2> and <U2, R2, F2, Um2, Rm2, Fm2>. Their permutations lead to exactly the same cube positions, even though the puzzle is held in a different orientation.

Some of the groups on this list are actually identical. Any two identical groups on the list can be proved to be identical with one or more of the following move sequences.

 1 Rm = U Rm2 Um Fm2 U Fm U' Fm2 Um' Rm2 U' ( uses U, Um, Rm2, Fm ) 2 R = U F R2 U Um F U'Um' F' R2 U' ( uses U, F, R2, Um ) 3 U = R2 F2 U2 R F2 Um R2 Um2 F' R2 Um' F2 Um2 R' U2 F2 R2 ( uses U2, F, R, Um ) 4 D = R2 Rm2 U2 D2 R2 U R2 D2 U2 Rm2 R2 ( uses U, R2, Rm2, D2 )

Middle layer turns do not affect the corners so you can solve corners using the same sequences that were used in the previous section. The centres are then completely trivial to solve next, though sometimes it is easier to solve the edges first and only then do the centres. Here are some move sequences that may be useful. I think that with conjugates of these, all positions in the above groups can be solved.

DF+ DB+ UB+ UL+U, RmRm U Rm U Rm U Rm U
UF+ UB+U, RmRm U Rm U Rm U2 Rm' U Rm' U Rm' U2
UF+ UR+U, RmRm2 U Rm U2 Rm' U Rm' U Rm' U2 Rm U Rm'
(DF, UF, UB)U2, RmU2 Rm U2 Rm'
(UF, UB)(DF, DB) Two H patternU2, Rm2U2 Rm2 U2 Rm2
(UF, DB)(DF, UB) Four H patternU2, F2, Rm2U2 Rm2 U2 F2 Rm2 F2
(U,R,F)(D,L,B) Six-spotRm, UmUm' Rm' Um Rm
(R,L)(F,B) Four-spotRm2, UmUm' Rm2 Um Rm2

## Subgroups generated by 'wide' moves.

Now we look at the groups that occur when a 'wide' move is used, which is a move of two adjacent layers simultaneously. For example Uw is a move equivalent to Um U. In the table below the generators for each group will include one or more wide moves, possibly combined with normal or middle layer moves. As before I will only list those groups for which there is at least one corner or edge that remains in place that can serve as a reference point when solving the puzzle. Also, on this list a wide move will never be combined with a quarter turn of one of its two layers because that is equivalent to just using the two layers separately. Furthermore, Uw Um2 is equal to Uw' U2 so the only remaining non-trivial combination of wide move and parallel layer move Uw U2. This leads to 298 more ways to generate a group.

Skip past table
GeneratorsFixed pieces SizeCornersEdgesCentresRestrictions
1a. Uw2 D layer 2= 2* 2* 2/ 4
1b. Uw D layer 4= 4* 4* 4/16
2a. Uw, U2 D layer 8= 4* 2 4* 4/16
2c. Uw2, R2 DBL DL DFL DF DB D 24= 3!* 8 3! / 2* 2/12
2d. Uw, R2 DBL DL DFL DF DB D 345,600= 6!/6* 5! 4!* 4/ 4
2e. Uw2, R DBL DL DFL DF DB D 604,800= 6!/6* 2 7!* 2/ 4
2f. Uw, R DBL DL DFL DF DB D 5,417,769,369,600= 6!/6 36/3* 9! 29/2* 4/ 2
2g. Uw2, Rw2 DBL DL DFL 24= 3!* 8 3! / 2*2 2/24
2h. Uw, Rw2 DBL DL DFL 14,515,200= 6!/6* 4! 7!*2 4/ 8
2i. Uw, Rw DBL DL DFL 7,151,455,567,872,000= 6!/6 36/3* 11!/2 211/2*24/ 2
2j. Uw2, Rm2 DBL DL DFL DBR DR DFR 8= 2* 8*2 2/ 8
2k. Uw2, Rm DBL DL DFL DBR DR DFR 96= 2* 2 4!*2 4/ 8
2l. Uw, Rm2 DBL DL DFL DBR DR DFR 192= 4* 4 3! 23 / 2*2 4/16
2m. Uw, Rm DBL DL DFL DBR DR DFR 1,105,920= 4* 4 6! 26/2*24/ 8
3a. Uw, U2, R2 DBL DL DFL DF DB D 345,600= 6!/6* 5! 4!* 4/ 4
3b. Uw, U2, R DBL DL DFL DF DB D 5,417,769,369,600= 6!/6 36/3* 9! 29/2* 4/ 2
3e. Uw, U2, Rm2 DBL DL DFL DBR DR DFR 384= 4* 4 3! 23 / 2*2 4/8
3f. Uw, U2, Rm DBL DL DFL DBR DR DFR 2,211,840= 4* 4 6! 26/2*24/4
3i. Uw, U2, Rw2 DBL DL DFL 14,515,200= 6!/6* 4! 7!*2 4/ 8
3j. Uw, U2, Rw DBL DL DFL 7,151,455,567,872,000= 6!/6 36/3* 11!/2 211/2*24/ 2
3m. Uw2, R2, L2 DF DB D 96= 4!* 8 4! / 2* 2/48
3n. Uw, R2, L2 DF DB D 696,729,600= 8!* 4! 6!* 4/ 4
3o. Uw2, R, L2 DF DB D 1,625,702,400= 8!* 2 8!* 2/ 4
3p. Uw, R, L2 DF DB D 327,666,691,473,408,000= 8! 38/3* 10! 210/2* 4/ 2
3q. Uw2, R, L DF DB D 1,625,702,400= 8!* 2 8!* 2/ 4
3r. Uw, R, L DF DB D 327,666,691,473,408,000= 8! 38/3* 10! 210/2* 4/ 2
3s. Uw2, R2, F2 DBL DL DB D 10,368= 4!* 4! 3! 3! / 2* 2/ 2
3t. Uw, R2, F2 DBL DL DB D 87,091,200= 7!* 4! 6!* 4/ 4
3u. Uw2, R, F2 DBL DL DB D 152,409,600= 7!* 7! 3!* 2/ 2
3v. Uw, R, F2 DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
3w. Uw2, R, F DBL DL DB D 13,332,791,808,000= 7! 37/3* 10!* 2/ 2
3x. Uw, R, F DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
3y. Uw2, R2, Rm2 DBL DL DFL 96= 3!* 3! 4 8 /2 *2 2/24
3z. Uw, R2, Rm2 DBL DL DFL 29,030,400= 6!/6* 4! 7!*2 4/ 4
3aa.Uw2, R, Rm2 DBL DL DFL 2,419,200= 6!/6* 8!*2 2/ 8
3ab.Uw2, R2, Rm DBL DL DFL 1,152= 3!* 4 3! 4!*2 4/24
3ac.Uw, R, Rm2 DBL DL DFL 4,767,637,045,248,000= 6!/6 36/3* 11! 211/2*2 4/ 2
3ad.Uw, R2, Rm DBL DL DFL 11,147,673,600= 6!/6* 4! 2 7! 27/2*24/ 4
3ae.Uw2, R, Rm DBL DL DFL 29,030,400= 6!/6* 2 4! 7!*2 4/ 8
3af.Uw, R, Rm DBL DL DFL 14,302,911,135,744,000= 6!/6 36/3* 11! 211/2*24/ 2
3ag.Uw2, R2, Fm2 DBL DB DFL DF 192= 3!* 4! 8 / 2*2 2/12
3ah.Uw, R2, Fm2 DBL DB DFL DF 4,147,200= 6!/6* 6! 4!*2 4/ 4
3ai.Uw2, R, Fm2 DBL DB DFL DF 9,676,800= 6!/6* 2 8!*2 2/ 4
3aj.Uw2, R2, Fm DBL DB DFL DF 1,152= 3!* 8 4!*2 4/ 8
3ak.Uw, R, Fm2 DBL DB DFL DF 216,710,774,784,000= 6!/6 36/3* 10! 210/2*2 4/ 2
3al.Uw, R2, Fm DBL DB DFL DF 796,262,400= 6!/6* 2 4! 6! 26/2*24/ 4
3am.Uw2, R, Fm DBL DB DFL DF 2,477,260,800= 6!/6* 2 8! 28/2*2 4/ 4
3an.Uw, R, Fm DBL DB DFL DF 650,132,324,352,000= 6!/6 36/3* 10! 210/2*24/ 2
3ao.Uw2, Rm2, Fm2 DBL DBR DFL DFR 32= 2* 8 8 / 2*2 2 2 / 2/ 8
3ap.Uw, Rm2, Fm2 DBL DBR DFL DFR 1,536= 4* 4 4! 24/2*2 4/16
3aq.Uw2, Rm, Fm2 DBL DBR DFL DFR 384= 2* 8 4!*2 4/ 8
3ar.Uw, Rm, Fm2 DBL DBR DFL DFR 247,726,080= 4* 4 8! 28/2*24/ 8
3as.Uw2, Rm, Fm DBL DBR DFL DFR 13,824= 2* 2 4! 4!*24/ 4
3at.Uw, Rm, Fm DBL DBR DFL DFR 247,726,080= 4* 4 8! 28/2*24/ 8
3au.Uw2, Rw2, F2 DBL DL 41,472= 4!* 3! 4! 4! / 2*2 2/ 4
3av.Uw, Rw2, F2 DBL DL 1,219,276,800= 7!* 4! 7!*2 4/ 4
3aw.Uw2, Rw2, F DBL DL 1,219,276,800= 7!* 3! 8!*2 2/ 4
3ax.Uw, Rw, F2 DBL DL 901,083,401,551,872,000= 7! 37/3* 11!/2 211/2*24/ 2
3ay.Uw, Rw2, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
3az.Uw, Rw, F DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
3ba.Uw2, Rw2, Fm2 DBL DFL 192= 3!* 2! 2! 4!*2 2 2 / 2/12
3bb.Uw, Rw2, Fm2 DBL DFL 232,243,200= 6!/6* 4! 8!*2 4/ 4
3bc.Uw2, Rw2, Fm DBL DFL 1,152= 3!* 2! 2! 2 4!*2 4/ 8
3bd.Uw, Rw, Fm2 DBL DFL 171,634,933,628,928,000= 6!/6 36/3* 12!/2 212/2*24/ 2
3be.Uw, Rw2, Fm DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2*24/ 4
3bf.Uw, Rw, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24/ 2
3bg.Uw2, Rw2, Fw2 DBL 165,888= 4!* 4! 4! 4!/2*2 2 2 / 2/ 4
3bh.Uw, Rw2, Fw2 DBL 4,877,107,200= 7!* 4! 8!*2 4/ 8
3bi.Uw, Rw, Fw2 DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2*24/ 2
3bj.Uw, Rw, Fw DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2*24/ 2
4a. Uw, U2, R2, F2 DBL DL DB D 87,091,200= 7!* 4! 6!* 4/ 4
4b. Uw, U2, R2, F DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
4c. Uw, U2, R, F DBL DL DB D 13,652,778,811,392,000= 7! 37/3* 10! 210/2* 4/ 2
4g. Uw, U2, R2, L2 DF DB D 696,729,600= 8!* 4! 6!* 4/ 4
4h. Uw, U2, R, L2 DF DB D 327,666,691,473,408,000= 8! 37/3* 10! 210/2* 4/ 2
4i. Uw, U2, R, L DF DB D 327,666,691,473,408,000= 8! 37/3* 10! 210/2* 4/ 2
4m. Uw, U2, Rm2, F2 DBL DL DBR BR 4,147,200= 6!/6* 4! 6!* 8/ 4
4n. Uw, U2, Rm2, F DBL DL DBR BR 216,710,774,784,000= 6!/6 36/3* 10! 210/2* 8/ 2
4o. Uw, U2, Rm, F2 DBL DL DBR BR 796,262,400= 6!/6* 2 4! 6! 26/2* 24/ 4
4p. Uw, U2, Rm, F DBL DL DBR BR 650,132,324,352,000= 6!/6 36/3* 10! 210/2* 24/ 2
4u. Uw, U2, Rm2, R2 DBL DL DFL 29,030,400= 6!/6* 4! 7!* 8/ 4
4v. Uw, U2, Rm2, R DBL DL DFL 4,767,637,045,248,000= 6!/6 36/3* 11! 211/2* 8/ 2
4w. Uw, U2, Rm, R2 DBL DL DFL 11,147,673,600= 6!/6* 2 4! 7! 27/2* 24/ 4
4x. Uw, U2, Rm, R DBL DL DFL 14,302,911,135,744,000= 6!/6 36/3* 11! 211/2* 24/ 2
4ae.Uw, U2, Rw, R2 DBL DL DFL 7,151,455,567,872,000= 6!/6 36/3* 11!/2 211/2* 24/ 2
4af.Uw, U2, Rm2, Fm2 DBL DBR DFL DFR 3,072= 4* 4 4! 24/2* 8/ 8
4ag.Uw, U2, Rm2, Fm DBL DBR DFL DFR 495,452,160= 4* 4 8! 28/2* 24/ 4
4ah.Uw, U2, Rm, Fm DBL DBR DFL DFR 495,452,160= 4* 4 8! 28/2* 24/ 4
4al.Uw, U2, Rw2, F2 DBL DL 1,219,276,800= 7!* 4! 7!* 8/ 4
4am.Uw, U2, Rw2, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2* 8/ 2
4an.Uw, U2, Rw, F2 DBL DL 901,083,401,551,872,000= 7! 37/3* 11!/2 211/2* 24/ 2
4ao.Uw, U2, Rw, F DBL DL 180,216,680,310,374,4000= 7! 37/3* 11! 211/2* 24/ 2
4at.Uw, U2, Rw2, Fm2 DBL DFL 232,243,200= 6!/6* 4! 8!* 8/ 4
4au.Uw, U2, Rw2, Fm DBL DFL 178,362,777,600= 6!/6* 2 4! 8! 28/2* 24/ 4
4av.Uw, U2, Rw, Fm2 DBL DFL 171,634,933,628,928,000= 6!/6 36/3* 12!/2 212/2* 24/ 2
4aw.Uw, U2, Rw, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2* 24/ 2
4bb.Uw, U2, Rw2, Fw2 DBL 4,877,107,200= 7!* 4! 8!* 8/ 8
4bc.Uw, U2, Rw2, Fw DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
4bd.Uw, U2, Rw, Fw DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
4a.Uw2, R2, F2, L2 D DB 165,888= 4! 4!/6* 3! 4! 4! / 2* 2/ 2
4b.Uw, R2, F2, L2 D DB 4,877,107,200= 8!* 4! 7!* 4/ 4
4c.Uw2, R, F2, L2 D DB 9,754,214,400= 8!* 3! 8!* 2/ 2
4d.Uw2, R2, F, L2 D DB 4,877,107,200= 8!* 4! 7!* 2/ 2
4e.Uw, R, F2, L2 D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
4f.Uw, R2, F, L2 D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
4g.Uw2, R, F, L2 D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
4h.Uw2, R, F2, L D DB 9,754,214,400= 8!* 3! 8!* 2/ 2
4i.Uw, R, F, L2 D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
4j.Uw, R, F2, L D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
4k.Uw2, R, F, L D DB 3,519,857,037,312,000= 8! 38/3* 11!* 2/ 2
4l.Uw, R, F, L D DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
4m.Uw2, Rm2, R2, F2 DBL DL 82,944= 4!* 3! 4! 4! / 2*2 2/ 2
4n.Uw, Rm2, R2, F2 DBL DL 1,219,276,800= 7!* 4! 7!*2 4/ 4
4o.Uw2, Rm, R2, F2 DBL DL 165,888= 4!* 3! 4! 4!*2 4/ 4
4p.Uw2, Rm2, R, F2 DBL DL 1,219,276,800= 7!* 4! 7!*2 2/ 2
4q.Uw2, Rm2, R2, F DBL DL 2,438,553,600= 7!* 3! 8!*2 2/ 2
4r.Uw, Rm, R2, F2 DBL DL 468,202,291,200= 7!* 4! 2 7! 27/2*24/ 4
4s.Uw, Rm2, R, F2 DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
4t.Uw2, Rm, R, F2 DBL DL 2,438,553,600= 7!* 2 4! 7!*2 4/ 4
4u.Uw, Rm2, R2, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
4v.Uw2, Rm, R2, F DBL DL 624,269,721,600= 7!* 3! 8! 28/2*2 4/ 2
4w.Uw2, Rm2, R, F DBL DL 293,321,419,776,000= 7! 37/3* 11!*2 2/ 2
4x.Uw, Rm, R, F2 DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
4y.Uw, Rm, R2, F DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
4z.Uw, Rm2, R, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
4aa.Uw2, Rm, R, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
4ab.Uw, Rm, R, F DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2*24/ 2
4ac.Uw2, Rm2, F2, B2 DL DR 768= 4!* 2! 2! 2! 4! / 2*2 2/12
4ad.Uw, Rm2, F2, B2 DL DR 1,393,459,200= 8!* 6! 4!*2 4/ 4
4ae.Uw2, Rm, F2, B2 DL DR 4,608= 4!* 2! 2! 2! 4!*2 4/ 8
4af.Uw2, Rm2, F, B2 DL DR 3,251,404,800= 8!* 2 8!*2 2/ 4
4ag.Uw, Rm, F2, B2 DL DR 267,544,166,400= 8!* 2 4! 6! 26/2*24/ 4
4ah.Uw, Rm2, F, B2 DL DR 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
4ai.Uw2, Rm, F, B2 DL DR 832,359,628,800= 8!* 2 8! 28/2*2 4/ 4
4aj.Uw2, Rm2, F, B DL DR 3,251,404,800= 8!* 2 8!*2 2/ 4
4ak.Uw, Rm, F, B2 DL DR 1,966,000,148,840,448,000= 8! 38/3* 10! 210/2*24/ 2
4al.Uw, Rm2, F, B DL DR 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
4am.Uw2, Rm, F, B DL DR 832,359,628,800= 8!* 2 8! 28/2*2 4/ 4
4an.Uw, Rm, F, B DL DR 1,966,000,148,840,448,000= 8! 38/3* 10! 210/2*24/ 2
4ao.Uw2, Rm2, Fm2, R2 DBL DFL 768= 3!* 4! 8 4 / 2*2 2 2 / 2/12
4ap.Uw, Rm2, Fm2, R2 DBL DFL 232,243,200= 6!/6* 4! 8!*2 4/ 4
4aq.Uw2, Rm, Fm2, R2 DBL DFL 9,216= 3!* 4 4! 4!*2 4/12
4ar.Uw2, Rm2, Fm, R2 DBL DFL 4,608= 3!* 4! 8 8 / 2*2 4/ 8
4as.Uw2, Rm2, Fm2, R DBL DFL 38,707,200= 6!/6* 8 8!*2 2 2 / 2/ 4
4at.Uw, Rm, Fm2, R2 DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2*24/ 4
4au.Uw, Rm2, Fm, R2 DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2*24/ 4
4av.Uw2, Rm, Fm, R2 DBL DFL 165,888= 3!* 8 4! 4!*24/ 4
4aw.Uw, Rm2, Fm2, R DBL DFL 114,423,289,085,952,000= 6!/6 36/3* 12! 212/2*2 4/ 2
4ax.Uw2, Rm, Fm2, R DBL DFL 464,486,400= 6!/6* 2 4! 8!*2 4/ 4
4ay.Uw2, Rm2, Fm, R DBL DFL 9,909,043,200= 6!/6* 8 8! 28/2*2 4/ 4
4az.Uw, Rm, Fm, R2 DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2*24/ 4
4ba.Uw, Rm, Fm2, R DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24/ 2
4bb.Uw, Rm2, Fm, R DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24/ 2
4bc.Uw2, Rm, Fm, R DBL DFL 356,725,555,200= 6!/6* 4! 2 8! 28/2*24/ 2
4bd.Uw, Rm, Fm, R DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2*24/ 2
4be.Uw2, Rw2, F2, B2 DL 165,888= 4! 4!/6* 3! 4! 4! / 2*2 2/ 4
4bf.Uw, Rw2, F2, B2 DL 9,754,214,400= 8!* 4! 7!*2 4/ 4
4bg.Uw2, Rw2, F, B2 DL 9,754,214,400= 8!* 3! 8!*2 2/ 4
4bh.Uw, Rw, F2, B2 DL 21,626,001,637,244,928,000= 8! 38/3* 11!/2 211/2*24/ 2
4bi.Uw, Rw2, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
4bj.Uw2, Rw2, F, B DL 9,754,214,400= 8!* 3! 8!*2 2/ 4
4bk.Uw, Rw, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
4bl.Uw, Rw2, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
4bm.Uw, Rw, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
4bn.Uw2, Rw2, Fm2, F2 DBL 331,776= 4!* 4! 4! 4! / 2*2 2 2 / 2/ 2
4bo.Uw2, Rw2, Fm2, F DBL 4,877,107,200= 7!* 4! 8!*2 2 2 / 2/ 4
4bp.Uw2, Rw2, Fm, F2 DBL 663,552= 4!* 4! 4! 4!*2 4/ 4
4bq.Uw2, Rw, Fm2, F2 DBL 9,754,214,400= 7!* 2 4! 8! / 2*2 4/ 4
4br.Uw2, Rw2, Fm, F DBL 9,754,214,400= 7!* 2 4! 8!*2 4/ 8
4bs.Uw2, Rw, Fm2, F DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
4bt.Uw2, Rw, Fm, F2 DBL 7,491,236,659,200= 7!* 4! 2 8! 28/2* 24/ 4
4bu.Uw, Rw, Fm2, F2 DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
4bv.Uw2, Rw, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
4bw.Uw, Rw, Fm2, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
4bx.Uw, Rw, Fm, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
4by.Uw, Rw, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
5a. Uw, U2, R2, L2, F2 DB 4,877,107,200= 8!* 4! 7!* 4/ 4
5b. Uw, U2, R2, L2, F DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5c. Uw, U2, R, L2, F2 DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5d. Uw, U2, R, L2, F DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5e. Uw, U2, R, L, F2 DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5f. Uw, U2, R, L, F DB 7,208,667,212,414,976,000= 8! 38/3* 11! 211/2* 4/ 2
5g. Uw, U2, Rm2, F2, B2 DL DR 1,393,459,200= 8!* 4! 6!*2 4/ 4
5h. Uw, U2, Rm2, F, B2 DL DR 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
5i. Uw, U2, Rm, F2, B2 DL DR 267,544,166,400= 8!* 4! 2 6! 26/2* 24/ 4
5j. Uw, U2, Rm2, F, B DL DR 655,333,382,946,816,000= 8! 38/3* 10! 210/2*2 4/ 2
5k. Uw, U2, Rm, F, B2 DL DR 1,966,000,148,840,448,000= 8! 38/3* 10! 210/2* 24/ 2
5l. Uw, U2, Rm, F, B DL DR 1,966,000,148,840,448,000= 8! 38/3* 10! 210/2* 24/ 2
5m. Uw, U2, Rm2, R2, F2 DBL DL 1,219,276,800= 7!* 4! 7!*2 4/ 4
5n. Uw, U2, Rm2, R, F2 DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5o. Uw, U2, Rm2, R2, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5p. Uw, U2, Rm, R2, F2 DBL DL 468,202,291,200= 7!* 4! 2 7! 27/2* 24/ 4
5q. Uw, U2, Rm2, R, F DBL DL 600,722,267,701,248,000= 7! 37/3* 11! 211/2*2 4/ 2
5r. Uw, U2, Rm, R2, F DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2* 24/ 2
5s. Uw, U2, Rm, R, F2 DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2* 24/ 2
5t. Uw, U2, Rm, R, F DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2* 24/ 2
5u. Uw, U2, Rm2, R2, Fm2 DBL DFL 232,243,200= 6!/6* 4! 8!*2 4/ 4
5v. Uw, U2, Rm2, R2, Fm DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2* 24/ 4
5w. Uw, U2, Rm2, R, Fm2 DBL DFL 114,423,289,085,952,000= 6!/6 36/3* 12! 212/2*2 4/ 2
5x. Uw, U2, Rm, R2, Fm2 DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2* 24/ 4
5y. Uw, U2, Rm2, R, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2* 24/ 2
5z. Uw, U2, Rm, R2, Fm DBL DFL 178,362,777,600= 6!/6* 4! 2 8! 28/2* 24/ 4
5aa.Uw, U2, Rm, R, Fm2 DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2* 24/ 2
5ab.Uw, U2, Rm, R, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2* 24/ 2
5ac.Uw, U2, Rw2, F2, B2 DL 9,754,214,400= 8!* 4! 7!*2 4/ 4
5ad.Uw, U2, Rw2, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5ae.Uw, U2, Rw, F2, B2 DL 21,626,001,637,244,928,000= 8! 38/3* 11!/2 211/2* 24/ 2
5af.Uw, U2, Rw2, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5ag.Uw, U2, Rw, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
5ah.Uw, U2, Rw, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
5ai.Uw, U2, Rw2, Fm2, F2 DBL 9,754,214,400= 7!* 4! 8!*2 4/ 4
5aj.Uw, U2, Rw2, Fm, F2 DBL 7,491,236,659,200= 7!* 4! 2 8! 28/2* 24/ 4
5ak.Uw, U2, Rw2, Fm2, F DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
5al.Uw, U2, Rw, Fm2, F2 DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
5am.Uw, U2, Rw2, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
5an.Uw, U2, Rw, Fm2, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
5ao.Uw, U2, Rw, Fm, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
5ap.Uw, U2, Rw, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
5aq.Uw, U2, Rw, R2, F2 DBL DL 901,083,401,551,872,000= 7! 37/3* 11!/2 211/2* 24/ 2
5ar.Uw, U2, Rw, R2, F DBL DL 1,802,166,803,103,744,000= 7! 37/3* 11! 211/2* 24/ 2
5as.Uw, U2, Rw, R2, Fm2 DBL DFL 171,634,933,628,928,000= 6!/6 36/3* 12!/2 212/2* 24/ 2
5at.Uw, U2, Rw, R2, Fm DBL DFL 343,269,867,257,856,000= 6!/6 36/3* 12! 212/2* 24/ 2
5au.Uw, U2, Rw, R2, Fw2 DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
5av.Uw, U2, Rw, R2, Fw DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
5aw.Uw2, Rm2, R2, F2, B2 DL 331,776= 4! 4!/6* 3! 4! 4! / 2*2 2/ 2
5ax.Uw, Rm2, R2, F2, B2 DL 9,754,214,400= 8!* 4! 7!*2 4/ 4
5ay.Uw2, Rm, R2, F2, B2 DL 663,552= 4! 4!/6* 3! 4! 4!*2 4/ 4
5az.Uw2, Rm2, R, F2, B2 DL 9,754,214,400= 8!* 4! 7!*2 2/ 2
5ba.Uw2, Rm2, R2, F, B2 DL 19,508,428,800= 8!* 3! 8!*2 2/ 2
5bb.Uw, Rm, R2, F2, B2 DL 3,745,618,329,600= 8!* 4! 2 7! 27/2*24/ 4
5bc.Uw, Rm2, R, F2, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bd.Uw2, Rm, R, F2, B2 DL 19,508,428,800= 8!* 2 4! 7!*2 4/ 4
5be.Uw, Rm2, R2, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bf.Uw2, Rm, R2, F, B2 DL 4,994,157,772,800= 8!* 3! 8! 28/2*2 4/ 2
5bg.Uw2, Rm2, R, F, B2 DL 7,039,714,074,624,000= 8! 38/3* 11!*2 2/ 2
5bh.Uw2, Rm2, R2, F, B DL 19,508,428,800= 8!* 3! 8!*2 2/ 2
5bi.Uw, Rm, R, F2, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
5bj.Uw, Rm, R2, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
5bk.Uw, Rm2, R, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bl.Uw2, Rm, R, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bm.Uw, Rm2, R2, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bn.Uw2, Rm, R2, F, B DL 4,994,157,772,800= 8!* 3! 8! 28/2*2 4/ 2
5bo.Uw2, Rm2, R, F, B DL 7,039,714,074,624,000= 8! 38/3* 11!*2 2/ 2
5bp.Uw, Rm, R, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
5bq.Uw, Rm, R2, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
5br.Uw, Rm2, R, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bs.Uw2, Rm, R, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
5bt.Uw, Rm, R, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2*24/ 2
5bu.Uw2, Rm2, Fm2, R2, F2 DBL 663,552= 4!* 4! 4! 4!/2*2 2 2 / 2/ 1
5bv.Uw, Rm2, Fm2, R2, F2 DBL 9,754,214,400= 7!* 4! 8!*2 4/ 4
5ca.Uw2, Rm, Fm2, R2, F2 DBL 1,327,104= 4!* 4! 4! 4!*2 4/ 2
5cb.Uw2, Rm2, Fm2, R, F2 DBL 9,754,214,400= 7!* 4! 8!*2 2 2 / 2/ 2
5cc.Uw, Rm, Fm2, R2, F2 DBL 7,491,236,659,200= 7!* 4! 2 8! 28/2*24/ 4
5cd.Uw, Rm2, Fm2, R, F2 DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
5ce.Uw2, Rm, Fm, R2, F2 DBL 7,962,624= 4!* 2 4! 4! 4!*24/ 2
5cf.Uw2, Rm, Fm2, R, F2 DBL 19,508,428,800= 7!* 2 4! 8!*2 4/ 4
5cg.Uw2, Rm2, Fm, R, F2 DBL 2,497,078,886,400= 7!* 4! 8! 28/2*2 4/ 2
5ch.Uw2, Rm2, Fm2, R, F DBL 3,519,857,037,312,000= 7! 37/3* 12!*2 2 2 / 2/ 2
5ci.Uw, Rm, Fm, R2, F2 DBL 7,491,236,659,200= 7!* 4! 2 8! 28/2*24/ 4
5cj.Uw, Rm, Fm2, R, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
5ck.Uw, Rm2, Fm, R, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
5cl.Uw, Rm2, Fm2, R, F DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
5cm.Uw2, Rm, Fm, R, F2 DBL 14,982,473,318,400= 7!* 4! 2 8! 28/2*24/ 2
5cn.Uw2, Rm, Fm2, R, F DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
5co.Uw, Rm, Fm, R, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
5cp.Uw, Rm, Fm2, R, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
5cq.Uw2, Rm, Fm, R, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
5cr.Uw, Rm, Fm, R, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2*24/ 2
6a. Uw, U2, Rm2, R2, F2, B2 DL 9,754,214,400= 8!* 4! 7!*2 4/ 4
6b. Uw, U2, Rm, R2, F2, B2 DL 3,745,618,329,600= 8!* 4! 2 7! 27/2* 24/ 4
6c. Uw, U2, Rm2, R2, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6d. Uw, U2, Rm2, R, F2, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6e. Uw, U2, Rm, R2, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6f. Uw, U2, Rm, R, F2, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6g. Uw, U2, Rm2, R2, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6h. Uw, U2, Rm2, R, F, B2 DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6i. Uw, U2, Rm, R2, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6j. Uw, U2, Rm, R, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6k. Uw, U2, Rm2, R, F, B DL 14,417,334,424,829,952,000= 8! 38/3* 11! 211/2*2 4/ 2
6l. Uw, U2, Rm, R, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6m. Uw, U2, Rm2, R2, Fm2, F2DBL 9,754,214,400= 7!* 4! 8!*2 4/ 4
6n. Uw, U2, Rm2, R2, Fm, F2 DBL 7,491,236,659,200= 7!* 4! 2 8! 28/2* 24/ 4
6o. Uw, U2, Rm2, R2, Fm2, F DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
6p. Uw, U2, Rm, R2, Fm, F2 DBL 7,491,236,659,200= 7!* 4! 2 8! 28/2* 24/ 4
6q. Uw, U2, Rm2, R, Fm, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6r. Uw, U2, Rm2, R, Fm2, F DBL 14,417,334,424,829,952,000= 7! 37/3* 12! 212/2*2 4/ 2
6s. Uw, U2, Rm2, R2, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6t. Uw, U2, Rm, R2, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6u. Uw, U2, Rm2, R, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6v. Uw, U2, Rm, R, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6w. Uw, U2, Rw, R2, F2, B2 DL 21,626,001,637,244,928,000= 8! 38/3* 11!/2 211/2* 24/ 2
6x. Uw, U2, Rw, R2, F, B2 DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6y. Uw, U2, Rw, R2, F, B DL 43,252,003,274,489,856,000= 8! 38/3* 11! 211/2* 24/ 2
6z. Uw, U2, Rw, R2, Fm2, F2 DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2
6aa.Uw, U2, Rw, R2, Fm, F2 DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6ab.Uw, U2, Rw, R2, Fm2, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6ac.Uw, U2, Rw, R2, Fm, F DBL 43,252,003,274,489,856,000= 7! 37/3* 12! 212/2* 24/ 2
6ad.Uw, U2, Rw, R2, Fw, F2 DBL 21,626,001,637,244,928,000= 7! 37/3* 12!/2 212/2* 24/ 2

Again many of the listed groups are actually identical. Any two identical groups on the list can be proved to be identical with one or more of the following move sequences.

 1 U (Um') = R2 Uw2 R2 Uw' R' Uw' R Uw' R Uw R' Uw2 R' Uw2 R' Uw R Uw' R Uw' R' Uw' R2 Uw2 R2 (Uw') ( uses Uw, R ) 2 U2 (Um2) = R2 Uw R2 Uw2 R2 Uw R2 Uw2 R2 Uw R2 Uw2 R2 Uw R2 Uw2 R2 Uw' R2 Uw2 R2 Uw2 R2 Uw2 R2 Uw' R2 (Uw2) ( uses Uw, R2 ) 3 U2 (Um2) = Rw2 Uw Rw2 Uw2 Rw2 Uw Rw2 Uw Rw2 Uw' Rw2 Uw2 Rw2 Uw' Rw2 Uw2 Rw2 Uw Rw2 Uw2 Rw2 Uw Rw2 Uw Rw2 Uw' Rw2 Uw2 Rw2 Uw' Rw2 (Uw2) ( uses Uw, Rw2 ) 4 U2 (Um2) = R2 F2 Uw2 R F2 R2 F2 R2 F2 R' Uw2 F2 R2 (Uw2) ( uses Uw2, R, F2 ) 5 Um2 (U2) = Rw Fm2 Rw Uw2 Rw2 Uw2 Rw2 Uw2 Rw' Fm2 Rw Uw2 Rw2 Uw2 Rw2 Uw2 Rw' Fm2 Rw' (Uw2) ( uses Uw2, Rw, Fm2 ) 6 U2 (Um2) = Rw2 Uw2 Rw' F2 Rw2 F2 Rw2 F2 Rw Uw2 Rw2 (Uw2) ( uses Uw2, Rw, F2 ) 7 Um2 (U2) = Rm2 R2 F2 Rm2 Uw2 F2 Uw2 F2 Rm2 Uw2 F2 Uw2 Fm2 F2 Uw2 Fm2 F2 Uw2 R2 Fm2 (Uw2) ( uses Uw2, Rm2, R2, Fm2, F2 ) 8 Um (U') = Fm2 Rw' Fm' Uw2 Fm Rw Fm2 Rw' Fm' Rw Fm2 Rw' Fm' Uw2 Fm Rw Fm2 (Uw') ( uses Uw, Rw, Fm ) 9 L = Uw2 L2 Uw2 R2 Uw2 R Uw2 R2 Uw2 L2 Uw2 ( uses Uw2, R, L2 ) 10 F = Uw2 F2 Uw' R2 Uw2 R Uw2 R2 Uw F2 Uw2 ( uses Uw, R, F2 ) 11 Fm = Uw' Fm2 Uw' Rm2 Uw Rm Uw' Rm2 Uw Fm2 Uw ( uses Uw, Rm, Fm2 ) 12 Fw = Uw Rw Uw2 Rw' Fw2 Rw' Fw2 Uw' Rw' Uw' Rw Uw Rw' Uw Rw Uw Rw' Uw' Rw Uw Rw Uw' Rw' Uw' Rw Uw' Rw' Uw Rw Uw Fw2 Rw Fw2 Rw Uw2 Rw' Uw' ( uses Uw, Rw, Fw2 )

All these sequences are optimal (if half turns count as 1 move) except for the last. I tried to get my computer to find an optimal solution for it, and after about three days it had reached depth 28 and still found nothing. If you can find a better solution than the 37 moves used here, let me know.

Solving a cube position within one of these groups tends to be rather difficult, since the wide turns affect so many pieces. Here are some move sequences involving wide turns that may be useful. I think that with conjugates of these and the previous move sequences, all positions in the above groups can be solved. These are all optimal.

(DF, UF, UB)Uw, RwUw Rw Uw' Rw Uw2 Rw Uw' Rw' Uw Rw' Uw' Rw Uw2 Rw Uw' Rw' Uw Rw2 Uw'
(UR, UF, UB)Uw, RwRw2 Uw2 Rw2 Uw2 Rw' Uw' Rw Uw2 Rw Uw' Rw' Uw Rw' Uw' Rw Uw2 Rw Uw' Rw' Uw' Rw2 Uw2 Rw2
UR+, UB+Uw, RwRw' Uw Rw' Uw' Rw2 Uw2 Rw Uw2 Rw' Uw2 Rw' Uw' Rw Uw2 Rw Uw' Rw Uw Rw2 Uw2 Rw' Uw2 Rw Uw2 Rw Uw Rw' Uw2
UF+, UB+Uw, RwRw2 Uw Rw' Uw' Rw2 Uw Rw Uw2 Rw Uw2 Rw' Uw' Rw' Uw Rw' Uw2 Rw Uw' Rw' Uw2 Rw Uw Rw2 Uw Rw2 Uw' Rw' Uw' Rw Uw'
(U,L,F)(D,R,B) Six-spotUw, RwRw Uw2 Rw Uw' Rw' Uw2 Rw' Uw2 Rw Uw Rw Uw Rw2 Uw' Rw2 Uw' Rw Uw Rw2 Uw Rw2 Uw' Rw' Uw' Rw' Uw2
(R,L)(F,B) Four-spotUw, RwRw Uw' Rw2 Uw' Rw Uw2 Rw Uw' Rw2 Uw' Rw Uw2 Rw Uw' Rw2 Uw' Rw Uw2
(UR, UF, UB)Uw, Rw2Rw2 Uw2 Rw2 Uw Rw2 Uw' Rw2 Uw2 Rw2 Uw2 Rw2 Uw' Rw2 Uw' Rw2 Uw2 Rw2 Uw Rw2 Uw' Rw2 Uw2 Rw2 Uw2 Rw2 Uw' Rw2 Uw'
(DF, UF, UB)Uw, Rw2Rw2 Uw2 Rw2 Uw Rw2 Uw' Rw2 Uw2 Rw2 Uw Rw2 Uw Rw2 Uw2 Rw2 Uw2 Rw2 Uw Rw2 Uw' Rw2 Uw2 Rw2 Uw Rw2 Uw Rw2 Uw2
(R,L)(F,B) Four-spotUw, Rw2Rw2 Uw Rw2 Uw Rw2 Uw Rw2 Uw2 Rw2 Uw' Rw2 Uw' Rw2 Uw2 Rw2 Uw' Rw2 Uw Rw2 Uw' Rw2 Uw' Rw2 Uw' Rw2 Uw2 Rw2 Uw Rw2 Uw Rw2 Uw2 Rw2 Uw Rw2 Uw'
(DF, UF, UB)Uw2, RmRm2 Uw2 Rm' Uw2 Rm
UF+, UB+Uw, RmRm Uw Rm Uw Rm' Uw2 Rm' Uw' Rm Uw' Rm' Uw2 Rm2
(UR, UF, UB)Uw, R2Uw R2 Uw2 R2 Uw2 R2 Uw R2 Uw R2 Uw' R2 Uw2 R2 Uw2 R2 Uw' R2 Uw' R2
(RB, RF, LF)Uw, R2Uw R2 Uw R2 Uw' R2 Uw' R2 Uw R2 Uw R2 Uw' R2 Uw' R2
(UR, DR)(RF, RB)Uw, R2R2 Uw R2 Uw R2 Uw' R2 Uw R2 Uw2 R2 Uw R2 Uw' R2 Uw R2 Uw' R2 Uw2 R2 Uw2
(UF, UB)(RF, RB)Uw, R2Uw R2 Uw' R2 Uw' R2 Uw' R2 Uw2 R2 Uw' R2 Uw' R2 Uw' R2 Uw' R2 Uw2 R2 Uw2
(R,L)(F,B) Four-spotUw, R2(R2 Uw R2 Uw2)6
(RB, RF, RD)Uw2, RR2 Uw2 R' Uw2 R2 Uw2 R Uw2 R' Uw2 R' Uw2 R' Uw2 R Uw2
FL+ BR+Uw, RUw2 R Uw2 R Uw' R Uw2 R Uw2 R' Uw2 R' Uw R' Uw2 R'

## Subgroups generated by only slice/anti-slice moves

The groups generated by slice moves, or anti-slice moves are interesting because there are many nice patterns in these groups.

Generators SizeCornersEdgesRestrictions
a.U2D2, F2B2, R2L2 8= 4* 23/ 4
b.UD', FB', RL' 768= 24* 83/2 3!/2/24
c.UD, FB, RL 6,144=4·24*43 3! 23/2/24
d.UD, FB, RL, UD', FB', RL'15,925,248= 3! 4! 4* 4! 4! 4! 3! 23/2/ 6

Two of these, the slice-squared group and the slice group have already been listed using middle layer turns as 3ai and 3al in the second section. These two slice groups are easy to understand if you consider the corners as fixed in space and the centres as moving pieces. They are quite easy to solve. Just line up each slice of edges with the corners, and you will be left with at most a spot pattern.

The third on the list is the anti-slice group. Positions in this group are more difficult to solve. First solve the corners relative to each other, which is in essence similar to solving the corners in the square group 6a. Next orient the edges using the move sequence RL UD FB RL UD FB, which flips the edges in the U/D layers. You should now have a cube with only opposing colours on each face. Line up the centres with the corners using squared (anti-)slices. The edges can be solved using the sequence RL UD F2B2 UD RL which is a 4H pattern on the sides.

The last group on the list is generated by all slice and anti-slice moves. Note that combining a slice and an anti-slice move you get a half turn of a single face, e.g. RL RL' = R2, so the square group (6a) is a subgroup of this one. It is relatively easy to solve by bringing it to a position in the square group. The edge flip in the previous paragraph is useful for this.

## Various small subgroups

For many small finite groups, an isomorphic group can be found on the cube. For example a cyclic group of order 3 can be found by a 3-cycle of pieces. Below I will list some small groups and give generators on the cube that give such a group.

GroupSizeGenerator
C2 2 Any half turn such as R2, or any edge flip.
C3 3 Any 3-cycle such as R2UD'F2U'D, or R2F2R2F2, or any corner twist.
C4 4 Any face turn such as R
C2×C2 4 R2L2 and F2B2
C5 5 Any 5-cycle, for example D2R2D'R2L2UR'L'B2RL', or URU'R which is a pair of 5-cycles.
C6=C2×C3 6 R2F2
S3 3!=6 R2F2R2F2 and R2
C7 7 Any 7-cycle, or URU'F which is a pair of 7-cycles.
C8 8 Any 8-cycle (combined with a 2-cycle for parity reasons), e.g. R2F2 D2R2D2 B2R2 D R2L2 U F2B2,
or a flipped edge-4-cycle, e.g. L2D R2F2B2 L'R' F'L2F LR' F2B2U'.
The shortest is UDR2, which is an 8-cycle and 2-cycle of edges with two corner 4-cycles.
C2×C4 8 R2, L
C2×C2×C2 8 R2L2, F2B2, U2D2.
D4 8 UD', and FB R2L2 F'B'
Q, Quaternion group 8 i = (UR,UF)+(UL,UB)+ = R2 UR'U'R F R2 URU' F' UR2
j = (UB,UF)+(UR,UL)+ = L'U'B' U2 BLU FU2F'
k = (UL,UF)+(UB,UR)+ = B'U'R U2 FRF'R' U' BRU'R'
Note that they all have order 4 and that ijk=jki=kij=ii=jj=kk.
C11 11 Any 11-cycle of edges, for example (U R'L F D2)2.
A4 12 F2L2F2L2 and F2R2F2R2 gives all even permutations of the 4 corner columns.
C13 13 Impossible. 13 does not divide the order of the cube group.
C2×C2×C4 16 U2, D, Superflip.
C2×C2×C2×C2 16 U2D2, F2B2, R2L2, Superflip
D4×C2 16 UD', FB R2L2 F'B', and R2U2D2L2U2D2
S4 4!=24 L2 and R2L2F2R2F2 permute the 4 corner columns
C1260 1260 The highest possible order of an element of the cube group is 1260.
The shortest example is UR'UF'D2, and consists of a twisted 3-cycle and 5-cycle of corners, and a flipped 2-cycle and 7-cycle of edges.
C213 213=8192 The 6H pattern F2R2L2B2L2R2 in its 6 possible orientations,
the double corner swap U R2FR2 U2D2 L2BL2 UD2 in its 6 possible orientations,
and the 4-flip UF2D F2B2 DL2D F'B LB2L2 U'D F'R'L in its three possible orientations.